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Question #43935

Difficulty: 8/10 | 🟢 Free

The Question

An inductor of inductance $L$ and a semiconductor device are connected in series to an ideal DC voltage source of EMF $V = \frac{2}{9} I_0 R_0$ . Due to rapid thermal feedback, the dynamic resistance of the semiconductor device depends on the instantaneous current $i$ as $R(i) = R_0 \left(1 - \frac{i}{I_0}\right)$ , where $R_0$ and $I_0$ are positive constants. The circuit is closed at $t=0$ . What is the total work done by the voltage source from $t=0$ until the current reaches half of its steady-state value?

Options

A.
$\frac{2}{9} L I_0^2 \ln\left(\frac{9}{8}\right)$
B.
$\frac{1}{9} L I_0^2 \ln\left(\frac{3}{2}\right)$
C.
$\frac{4}{9} L I_0^2 \ln\left(\frac{4}{3}\right)$
D.
$\frac{2}{3} L I_0^2 \ln\left(\frac{9}{8}\right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Transient Current in RL Circuit|Dynamic Resistance (AC Resistance)|Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle non-linear transient circuits, specifically invalidating the standard $1-e^{-t/\tau}$ exponential growth model.

Students often fall into the trap of attempting to define an 'average' time constant or incorrectly applying the standard formula $W = \frac{1}{2}LI^2 + \int I^2R dt$ without accounting for the dynamic resistance $R(i)$ .

By coupling the thermal feedback into the resistance, the problem forces students to derive the governing non-linear differential equation from first principles and apply the Work-Energy theorem in its differential form ( $dW = Vi dt = Vi \frac{dt}{di} di$ ) to bypass the explicit time dependence.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Transient Current in RL Circuit
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free